ATOMIC MULTIPLET THEORY Atomic multiplet theory is the description of the atomic structure with quantum mechanics. The main concepts have been discussed in the lectures on optical spectroscopy. Here we repeat the basic aspects, which are needed for a general understanding of the concepts. We add spin-orbit coupling that is important for x-ray absorption. The starting point is the relativistic Schrödinger (or Dirac) equation of a single electron in an atom, HΨ=EΨ. In atoms where more than one electron is present there are four terms in the atomic Hamiltonian: The kinetic energy (HK), the interaction between the nucleus and the electrons (HN), the electron-electron repulsion (Hee) and the spin-orbit coupling of each electron (Hls). The total Hamiltonian is then given by: 2 2 2 pi −Ze e H = H + H + H + H = + + + (r ) l s ATOM K N ee ls 2m ri rij i i i N N pairs N The kinetic energy and the interaction with the nucleus are the same for all electrons in a given atomic configuration. They define the average energy of the configuration (Hav). The electron- electron repulsion and the spin-orbit coupling define the relative energy of the different terms within this configuration. We only use these two interactions to determine the relative energies of the possible configurations of the multi/electron states. 1 THE GROUND STATE OF HYDROGEN AND HELIUM We start with the hydrogen atom. A convenient way to describe the electronic nature of a state is to give its main quantum numbers. The ground state of the hydrogen atom has one electron in the 1s shell, i.e. 1s1. Graphical representation of the spin (left) and orbital moment (right). This state has a spin S of ½ that can be oriented south (ms=+ ½) or north (ms=- ½) and an angular moment L of 0. The term symbol is a shorthand way to describe the quantum numbers of a state, where they are given as 2S+1L. The term symbol of the ground state of a hydrogen atom is 2S. The ground state of the helium atom has two electrons in the 1s shell, i.e. 1s2. This state has a spin moment S of 0 as the two spins cancel each other. The angular moment L is also 0. The term symbol of the ground state of a hydrogen atom is 1S. The first excited state of helium contains an electron in the 1s orbital and a second electron in the 2s orbitals. This present a classic case of a 2-electron state. Each orbital contains an electron that can be either spin up or spin down, creating four different combinations of the two spins, respectively |½>|½>, |½>|-½>, |-½>|½> and |-½>|-½>. The additions of the two electrons yields three two-electron states with total spin momentum S= 1, respectively |1,1> = |½>|½>, |1,0> =1/√2( |½>|-½> + |-½>|½>) and |1,-1> = |-½>|-½>. IN addition there is one state with total spin momentum S=0, i.e. |0,0> =1/√2( |½>|-½> - |-½>|½>). The two- electron states are indicated as |1,1>, which means that its S quantum number is 1 and its MS quantum number is also 1, etc. Singlet and triplet states of the 1s2s configuration of the excited state of helium. 2 QUANTUM NUMBERS AND TERM SYMBOLS The overview of quantum numbers of a single electron and two electrons in an atom, as well as their nomenclature, is given in the table. The principal quantum number N is not important for the angular symmetry of a state. For a single electron, the quantum numbers are indicated with the orbital angular momentum L, the spin angular momentum S of ½, total angular momentum J, with two values L+1/2 and L-1/2, the magnetic quantum number ML, spin magnetic quantum number MS, and the total magnetic quantum number MJ. Values Values name symbol (single electron) (two electrons) Azimuthal Quantum Number L = 0 L = |L -L | L (min) (min) 1 2 Orbital Angular Momentum L(max)=N-1 L(max)=L1+L2 ML(min)=-L ML(min)=-L Magnetic Quantum Number ML ML(max)=L ML(max)=L Spin Quantum Number S S=½ S(min)= 0 S(max)=1 Spin Magnetic MS(min)=-S MS(min)=-S MS Quantum Number MS(max)=S MS(max)=S Total Angular Momentum J J(min)=L-S J(min)=L-S Quantum Number J(max)=L+S J(max)=L+S Total Magnetic MJ(min)=-J MJ(min)=-J MJ Quantum Number MJ(max)=J MJ(max)=J Overview of the quantum numbers and their nomenclature. The steps in between the minimum and maximum values is 1.0 in all cases. For a 2-electron configuration the maximum orbital angular momentum L is equal to the addition of the two individual orbital angular momenta, L1 and L2. The same rule applies to the spin angular momentum, implying that the spin angular momentum of two electrons can be either 1 or 0, and the total angular momentum J takes the values from |L-S| to L+S by the step 1. For multi-electron 2S+1 configurations with quantum numbers L, S, and J, in general, a term symbol is written as LJ, where the orbital angular momentum L is indicated with their familiar notation: S for L=0, P for L=1, etc. In the absence of spin-orbit coupling, all terms with the same L and S have the same energy, giving an energy level that is (2L+1)(2S+1)-fold degenerate. When spin-orbit coupling is important, the terms are split in energy according to their J-value with a degeneracy of 2J+1. The quantity 2S+1 is called the spin multiplicity of the term, and the terms are called singlet, doublet, triplet, quartet, etc. according to S = 0, 1/2, 1, 3/2, etc. A single s electron has an orbital angular momentum l=0, a spin angular momentum S=1/2 and a total angular momentum j=1/2. There is only one term 2 symbol S1/2. For one p electron, L=1, S=1/2, and J can be 1/2 or 3/2, corresponding to term 2 2 2 2 symbols P1/2 and P3/2. Similarly a single d electron has term symbols D3/2 and D5/2 and a single 2 2 f-electron F5/2 and F7/2. The degeneracy of these states is given by 2J+1. The general rules for adding two electrons are given by the addition of momenta. 3 ADDITION OF MOMENTA It is often required to add the momenta from two sources together to get states of definite total momentum. As an example, assume we are adding the orbital angular momentum from two electrons, L1 and L2 to get a total angular momentum J.. We will show that the total angular momentum quantum number takes on every value in the range We can understand this qualitatively in the vector model pictured below. We are adding two quantum vectors. Graphical representation of vector addition: l1 + l2 This addition is exactly the same for all momenta S, L and J, for example: • Add the orbital angular momentum to the spin angular momentum for an electron in an atom as J=L+S; • Add the orbital angular momenta together for two electrons in an atom L=L1+L2; • Add the spins of two particles together S=S1+S2; • Add the total angular momenta of two electrons together J=J1+J2 Essentially this rule governs all addition of momenta: Adding L1=3 with L2=2 yields as maximum Lmax =L1+L2=5 and as minimum Lmin = |L1-L2| = 1, with all possible values of L differing by an integer in-between. L can be 1, 2, 3, 4 or 5. It can always be checked if all values of L are found by adding the degeneracies of each state. The degeneracy of each moment is given as 2L+1. The degeneracy of L1=3 is 7 and the degeneracy of L2=2 is equal to 5. In total there are 7 times 5 is 35 possibilities. Adding the degeneracies of the final moments L yields ∑(2L+1) = 3+5+7+9+11 = 35. It is always useful to check the completeness of the result by calculating the total degeneracies. Exactly the same rules apply for adding S1 + S2, or L1 + S1 to yield the total moment J. 4 SPIN-ORBIT COUPLING The consequences of spin-orbit coupling have been described above, but the physical origin of spin-orbit coupling has not been treated yet. Spin-orbit coupling is a consequence of the inclusion of relativistic effects and it follows directly from the Dirac equation that can be considered as the relativistic Schrödinger equation. Classically one can make the following reasoning (that is not exactly correct, but it gives a good idea). Consider a hydrogen atom with the electron that orbits around the nucleus. Then one can also look at it as a proton orbiting the electron, i.e. the electron feels the electrostatic field of the proton but it also feels an effective current. This current produces a magnetic field that interacts with the magnetic moment of the electron. The spin-orbit coupling is graphically indicated in the figure below. The orbital moment of the electron (l) can be parallel or anti-parallel to the spin moment of the electron (s), yielding l+s respectively l-s. Graphical representation of spin-orbit coupling COUPLING SCHEMES WITH ONE ELECTRON PER SHELL Coupling schemes of states that have only one electron of one hole per shell can be calculated by applying the vector addition rules as described above.